An insurance company wants to examine the views of its clients about the prices of three car insurance plans launched last year. It conducts a survey with two sets of plans with different prices and finds that:

• If plan A is sold for \(\$ 150\), plan B for \(\$ 250\), and plan C for \(\$ 350\), then \(45 \%\) of the customers would be interested in plan A, \(15 \%\) in plan B, and \(40 \%\) in plan C.

• If plans A, B, and C are sold for \(\$ 170, \$ 250\), and \(\$ 310\) respectively, then \(15 \%\) of the customers would be interested in plan A, \(40 \%\) in plan B, and \(45 \%\) in plan C.

a. For the first pricing set, construct the probability distribution of \(X=\) selling price for the sale of a car insurance plan, find its mean, and interpret.

b. For the second pricing set, construct the probability distribution of \(X\), find its mean, and interpret.

c. Which pricing set is more profitable to the company? Explain.